# Prime matrices

Trip up to New England tomorrow. Be there.

Flying to New England for a while, so continuing with extra short posts. For today, here is a (I think) nice question I gave for my linear algebra class last summer, as a problem meant to encourage the use of technology:

It is clear that any 2 x 2 matrix, each of whose entries is a distinct prime, will be nonsingular.  For example, the matrix

$M = \begin{bmatrix} 2&5\\ 17&11 \end{bmatrix}$

is nonsingular.  However, there exist 3 x 3 matrices, each of whose entries is a distinct prime, which are singular.  For example, the matrix

$M=\begin{bmatrix}7&3&5\\23&11&13\\47&19&37\end{bmatrix}$

is singular.

Out of all such matrices (3 x 3, distinct prime entries, singular), which one has the smallest sum of entries? (where the solution will be this sum, since exchanging rows/columns will leave the determinant and sum invariant, but shows that the actual matrix will be nonunique)

Solution Friday.  As an aside, I have only upper bounds for the 4 x 4 case.

On the less computational side, do you have a proof that for any $n\ge 3$ there are infinitely many such singular matrices?